Sharp Sobolev inequalities with lower order remainder terms

Druet, Olivier; Hebey, Emmanuel; Vaugon, Michel

doi:10.1090/s0002-9947-00-02698-2

Cited by 19 publications

(16 citation statements)

References 23 publications

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“…The "local to global" approach has been systematically used by Aubin [3], Hebey and Vaugon [26], Aubin and Li [6], Druet, Hebey and Vaugon [19], and others. In [28,29], Li and Zhu introduced a global approach by attacking the problem directly on the whole manifold.…”

Section: Statement Of the Main Resultsmentioning

confidence: 99%

A sharp Sobolev inequality on Riemannian manifolds

Li¹,

Ricciardi

2002

Comptes Rendus. Mathématique

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Let (M, g) be a smooth compact Riemannian manifold without boundary of dimension n ≥ 6. We prove thatThe inequality is sharp in the sense that on any (M, g), K can not be replaced by any smaller number and Rg can not be replaced by any continuous function which is smaller than Rg at some point. If (M, g) is not locally conformally flat, the exponent 2n/(n + 2) can not be replaced by any smaller number. If (M, g) is locally conformally flat, a stronger inequality, with 2n/(n + 2) replaced by 1, holds in all dimensions n ≥ 3. key words: sharp Sobolev inequality, critical exponent, Yamabe problem MSC 2000 subject classification: 35J60, 58E35

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Section: Statement Of the Main Resultsmentioning

confidence: 99%

A sharp Sobolev inequality on Riemannian manifolds

Li¹,

Ricciardi

2002

Comptes Rendus. Mathématique

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“…Using now the system (19) on the first right-hand side integral and the Cauchy-Schwarz inequality on the second right-hand side integral, we deduce that…”

Section: Sharp Potential Type Sobolev Inequalitiesmentioning

confidence: 96%

“…As first pointed out by Brézis and Nirenberg in [10], an elliptic type situation may change drastically when passing from low dimensions, in this case n = 3, to high dimensions, say n ≥ 4. When dealing with sharp Sobolev inequalities on compact Riemannian manifolds, this phenomenon happened in the works of Hebey [22] and Druet, Hebey and Vaugon [19].…”

Section: On the Question (E)mentioning

confidence: 99%

Extremal maps in best constants vector theory. Part I: Duality and compactness

Barbosa

Montenegro

2012

Journal of Functional Analysis

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We develop a comprehensive study on sharp potential type Riemannian Sobolev inequalities of order 2 by means of a local geometric Sobolev inequality of same kind and suitable De Giorgi-Nash-Moser estimates. In particular we discuss questions like continuous dependence of optimal constants and existence and compactness of extremal maps. The main obstacle arising in the present setting lies at fairly weak conditions of regularity assumed on potential functions. Content

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“…The effect of the scalar curvature in the refined inequalities related to the Aubin conjecture was studied by Druet-Hebey-Vaugon [13], Druet-Hebey [12], Hebey [16], Li-Ricciardi [22] and etc. In particular, Li-Ricciardi [22] obtained an inequality involving the scalar curvature in the way of the mean curvature in (6), if n ≥ 6.…”

Section: Introductionmentioning

confidence: 99%